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We now turn to a more concrete example of a mathematical experiment. Our meta-goal in devising this experiment was to investigate the similarities and differences between experiments in mathematics and in the natural sciences, particularly in physics. We therefore resolved to examine a conjecture which could be approached by collecting and investigating a huge amount of data: the conjecture that every non-rational algebraic number is normal in every base (see box). It is important to understand that we did not aim to prove or disprove this conjecture; our aim was to find evidence pointing in one or the other direction. We were hoping to gain insight into the nature of the problem from an experimental perspective.

The actual experiment consisted of computing to 10,000 decimal digits the square roots and cube roots of the positive integers smaller than 1000 and then subjecting these data to certain statistical tests (again, see box). Under the hypothesis that the digits of these numbers are uniformly distributed (a much weaker hypothesis than normality of these numbers), we expected the probability values of the statistics to be distributed uniformly between 0 and 1. Our first run showed fairly conclusively that the digits were distributed uniformly. In fact, the Anderson-Darling test, which we used to measure how uniformly distributed our probabilities were suggested that the probabilities might have been `too uniform' to be random. We therefore ran the same tests again, only this time for the first 20,000 decimal digits, hoping to detect some non-randomness in the data. The data were not as interesting on the second run.

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